document.write( "Question 202197: Factor the following polynomial completely.\r
\n" ); document.write( "\n" ); document.write( "20x2 + 22xy + 6y2 = \n" ); document.write( "

Algebra.Com's Answer #152454 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"20x%5E2%2B22xy%2B6y%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%2810x%5E2%2B11xy%2B3y%5E2%29\"$ Factor out the GCF $\"2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"10x%5E2%2B11xy%2B3y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"10x%5E2%2B11xy%2B3y%5E2\"$ , we can see that the first coefficient is $\"10\"$ , the second coefficient is $\"11\"$ , and the last coefficient is $\"3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"10\"$ by the last coefficient $\"3\"$ to get $\"%2810%29%283%29=30\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"30\"$ (the previous product) and add to the second coefficient $\"11\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"30\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"30\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"30\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*30 = 30
\n" ); document.write( "2*15 = 30
\n" ); document.write( "3*10 = 30
\n" ); document.write( "5*6 = 30
\n" ); document.write( "(-1)*(-30) = 30
\n" ); document.write( "(-2)*(-15) = 30
\n" ); document.write( "(-3)*(-10) = 30
\n" ); document.write( "(-5)*(-6) = 30\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"11\"$ :\r
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First Number	Second Number	Sum
1	30	1+30=31
2	15	2+15=17
3	10	3+10=13
5	6	5+6=11
-1	-30	-1+(-30)=-31
-2	-15	-2+(-15)=-17
-3	-10	-3+(-10)=-13
-5	-6	-5+(-6)=-11

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"6\"$ add to $\"11\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"6\"$ both multiply to $\"30\"$ and add to $\"11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"11xy\"$ with $\"5xy%2B6xy\"$ . Remember, $\"5\"$ and $\"6\"$ add to $\"11\"$ . So this shows us that $\"5xy%2B6xy=11xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"10x%5E2%2Bhighlight%285xy%2B6xy%29%2B3y%5E2\"$ Replace the second term $\"11xy\"$ with $\"5xy%2B6xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2810x%5E2%2B5xy%29%2B%286xy%2B3y%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%282x%2By%29%2B%286xy%2B3y%5E2%29\"$ Factor out the GCF $\"5x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%282x%2By%29%2B3y%282x%2By%29\"$ Factor out $\"3y\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%2B3y%29%282x%2By%29\"$ Combine like terms. Or factor out the common term $\"2x%2By\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"2%2810x%5E2%2B11xy%2B3y%5E2%29\"$ then factors further to $\"2%285x%2B3y%29%282x%2By%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"20x%5E2%2B22xy%2B6y%5E2\"$ completely factors to $\"2%285x%2B3y%29%282x%2By%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"20x%5E2%2B22xy%2B6y%5E2=2%285x%2B3y%29%282x%2By%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2%285x%2B3y%29%282x%2By%29\"$ to get $\"20x%5E2%2B22xy%2B6y%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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